# A contact geometry framework for field theories with dissipation

## TL;DR

This paper introduces a novel geometric framework called $k$-contact structures for Hamiltonian field theories with dissipation, generalizing existing contact and $k$-symplectic formalisms, and applies it to examples like damped strings and Burgers' equation.

## Contribution

It develops the $k$-contact geometric framework for dissipative Hamiltonian field theories, extending contact geometry to field theories with dissipation.

## Key findings

- Defined $k$-contact structures and Hamiltonian systems with dissipation.
- Analyzed symmetries and dissipation laws within this framework.
- Applied the theory to damped vibrating string and Burgers' equation.

## Abstract

We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of $k$-contact structure and $k$-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the $k$-symplectic Hamiltonian systems in field theory. The concepts of symmetries and dissipation laws are introduced and developed. Two relevant examples are analyzed in detail: the damped vibrating string and Burgers' equation.